Particles and Fields

This volume focuses on quantum field theory: inegrable theories, statistical systems, and applications to condensed-matter physics. It covers some of the most significant recent advances in theoretical physics at a level accessible to advanced graduate students.

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The focus of this volume is on quantum field theory: inegrable theories, statistical systems, and applications to condensed-matter physics. It covers some of the most significant recent advances in theoretical physics at a level accessible to advanced graduate students. The contributions, each by a noted researcher, dicuss such topics as: some remarkable features of integrable Toda field theories (E. Corrigan), properties of a gas of interacting Fermions in a lattice of magnetic ions (J. Feldman &. al.), how quantum groups arise in three-dimensional topological quantum field thory (D. Freed), a method for computing correlation functions of solvable lattice models (T. Miwa), matrix models discussed from the point of view of integrable systems (A. Morozov), localization of path integrals in certain equivariant cohomologies (A. Niemi), Calogero-Moser systems (S. Ruijsenaars), planar gauge theories with broken symmetries (M. de Wild Propitius & F.A. Bais), quantum-Hall fluids (A. Capelli & al.), spectral theory of quantum vortex operators (P.I. Ettinghoff).

Contenu
1 Recent Developments in Affine Toda Quantum Field Theory.- 1 Introduction.- 2 Classical Integrability and Classical Data.- 3 Aspects of the Quantum Field Theory.- 4 Dual Pairs.- 5 A Word on Solitons.- 6 Other Matters.- 7 References.- 2 A Class of Fermi Liquids.- 1 Introduction.- 2 Four-Legged Diagrams.- 3 A Single-Slice Fermionic Cluster Expansion.- 4 References.- 3 Quantum Groups from Path Integrals.- 1 Classical Field Theory.- 2 Categories, Finite Groups, and Covering Spaces.- 3 Generalized Path Integrals.- 4 The Quantum Group.- 5 References.- 4 Half Transfer Matrices in Solvable Lattice Models.- 1 The Six-Vertex Model.- 2 The Antiferromagnetic Regime.- 3 Corner Transfer Matrix.- 4 Half Transfer Matrix.- 5 Commutation Relations.- 6 Correlation Functions.- 7 Two-Point Functions.- 8 Discussion.- 9 References.- 5 Matrix Models as Integrable Systems.- 1 Introduction.- 2 The Basic Example: Discrete 1-Matrix Model.- 3 Generalized Kontsevich Model.- 4 Kp/Toda ?-Function in Terms of Free Fermions.- 5 ?-Function as a Group-Theoretical Quantity.- 6 Conclusion.- 7 References.- 6 Localization, Equivariant Cohomology, and Integration Formulas 211.- 1 Symplectic Geometry.- 2 Equivariant Cohomology.- 3 Duistermaat-Heckman Integration Formula.- 4 Degeneracies.- 5 Equivariant Characteristic Classes.- 6 Loop Space.- 7 Example: Atiyah-Singer Index Theorem.- 8 Duistermaat-Heckman in Loop Space.- 9 General Integrable Models.- 10 Mathai-Quillen Formalism.- 11 Short Review of Morse Theory.- 12 Equivariant Mathai-Quillen Formalism.- 13 Equivariant Morse Theory.- 14 Loop Space and Morse Theory.- 15 Loop Space and Equivariant Morse Theory.- 16 Poincaré Supersymmetry and Equivariant Cohomology..- 17 References.- 7 Systems of Calogero-Moser Type.- 1 Introduction.- 2 Classical NonrelativisticCalogero-Moser and Toda Systems.- 3 Relativistic Versions at the Classical Level.- 4 Quantum Calogero-Moser and Toda Systems.- 5 Action-Angle Transforms.- 6 Eigenfunction Transforms.- 7 References.- 8 Discrete Gauge Theories.- 1 Broken Symmetry Revisited.- 2 Basics.- 3 Algebraic Structure.- 4 $${\overline D _2}$$ Gauge Theory.- 5 Concluding Remarks and Outlook.- 6 References.- 9 Quantum Hall Fluids as W1+? Minimal Models.- 1 Introduction.- 2 Dynamical Symmetry and Kinematics of Incompressible Fluids.- 3 Existing Theories of Edge Excitations and Experiments.- 4 W1+? Minimal Models.- 5 Further Developments.- 6 References.- 10 On the Spectral Theory of Quantum Vertex Operators 469 Pavel I. Etingof.- 1 Basic Definitions.- 2 Spectral Properties of Vertex Operators.- 3 The Semi-Infinite Tensor Product Construction.- 4 Computation of the Leading Eigenvalue and Eigenvector.- 5 References.